few_shots: Example:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K
options: Create equations that describe numbers or relationships

Gain familiarity with factors and multiples.

Use the four operations with whole numbers to solve problems.

Work with addition and subtraction equations.

Interpret the structure of expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Use properties of operations to generate equivalent expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Solve equations and inequalities in one variable.

Understand and apply properties of operations and the relationship between addition and subtraction.

Work with radicals and integer exponents.

Write and interpret numerical expressions.

Rewrite rational expressions.

Analyze patterns and relationships.

Solve systems of equations.

Perform arithmetic operations on polynomials.

Reason about and solve one-variable equations and inequalities.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Understand the relationship between zeros and factors of polynomials.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Represent and solve problems involving addition and subtraction.

Add and subtract within 20.

Use polynomial identities to solve problems.

Represent and solve problems involving multiplication and division.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with equal groups of objects to gain foundations for multiplication.

Understand the connections between proportional relationships, lines, and linear equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Write expressions in equivalent forms to solve problems.

Represent and solve equations and inequalities graphically.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Generate and analyze patterns.

Understand properties of multiplication and the relationship between multiplication and division.
problem_activity: Activity
In this activity, students write and graph equations to represent two constraints in the same situation and use tables and graphs to see possible values that satisfy the constraints. The work prompts them to think about a pair of values that simultaneously meets multiple constraints in the situation, which in turn helps them make sense of the phrase "a solution to both equations."
Monitor for these likely strategies for completing the tables, from less precise to more precise:
Guessing and checking
Using the graph (by identifying the point with a given
\(x\)
- or
\(y\)
-value and estimating the unknown value, or using technology to get an estimate)
Substituting the given value of one variable into the equation and solving for the other variable
Identify students who use each strategy and ask them to share later.
Launch
Arrange students in groups of 2 and provide access to graphing technology. Explain to students that they will now examine the same situation involving two quantities (raisins and walnuts), but there are now two constraints (cost and weight).
Give students 3–4 minutes of quiet time to complete the first set of questions about the cost constraint, and then pause for a whole-class discussion. Select previously identified students to share their strategies, in the order listed in the Activity Narrative. If one of the strategies is not mentioned, bring it up.
Ask a student who used the graph to complete the table to explain how exactly the graph was used. For example: "How did you use the graph of
\(4x+8y=15\)
to find
\(y\)
when
\(x\)
is 2?" Make sure students recognize that this typically involves hovering over different points on the graph or tracing the graph to get the coordinates.
Ask students to proceed with the remainder of the activity. If time permits, pause again after the second set of questions (about the weight constraint) to discuss how the strategies for completing the second table is like or unlike that of completing the first table. Alternatively, give students a minute to confer with their partner before moving on to the last question.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Here is a situation you saw earlier: Diego bought some raisins and walnuts to make trail mix. Raisins cost
$
4 a pound and walnuts cost
$
8 a pound. Diego spent
$
15 on both ingredients.
Write an equation to represent this constraint. Let
\(x\)
be the pounds of raisins and
\(y\)
be the pounds of walnuts.
Use graphing technology to graph the equation.
Complete the table with the amount of one ingredient Diego could have bought given the other. Be prepared to explain or show your reasoning.
| raisins (pounds) | walnuts (pounds) |
|------------------|------------------|
|        0         |                  |
|       0.25       |                  |
|                  |      1.375       |
|                  |       1.25       |
|       1.75       |                  |
|        3         |                  |
Here is a new piece of information: Diego bought a total of 2 pounds of raisins and walnuts combined.
Write an equation to represent this new constraint. Let
\(x\)
be the pounds of raisins and
\(y\)
be the pounds of walnuts.
Use graphing technology to graph the equation.
Complete the table with the amount of one ingredient Diego could have bought given the other. Be prepared to explain or show your reasoning.
| raisins (pounds) | walnuts (pounds) |
|------------------|------------------|
|        0         |                  |
|       0.25       |                  |
|                  |      1.375       |
|                  |       1.25       |
|       1.75       |                  |
|        3         |                  |
Diego spent
$
15 and bought exactly 2 pounds of raisins and walnuts. How many pounds of each did he buy? Explain or show how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students might think that the values in the second table need to reflect a total cost of
$
15. Clarify that the table represents only the constraint that Diego bought a total of 2 pounds of raisins and walnuts.
The idea of finding an
\((x,y)\)
pair that satisfies multiple constraints should be familiar from middle school. If students struggle to answer the last question, ask them to study the values in the table. Ask questions such as: "If Diego bought 0.25 pound of raisins, would he meet both the cost and weight requirements?" and "Which combinations of raisins and walnuts would allow him to meet both requirements? How many combinations are there?"
Activity Synthesis
Invite students to share their response and reasoning for the last question. Display the graphs representing the system (either created by a student or as shown in the Student Response). Discuss with students:
“Can you find other combinations of raisins and walnuts, besides 0.25 pound and 1.75 pounds, that meet both cost and weight constraints?” (No)
“How many possible combinations of raisins and walnuts meet both constraints? How do we know?” (One combination, because the graphs intersect only at one point.)
Explain to students that the two equations written to represent the constraints form a
system of equations
. We use a curly bracket to indicate a system, like this:
\(\begin{cases}\begin {align} 4x + 8y &= 15\\ x+ \hspace{2mm}y&=2 \end{align} \end{cases}\)
Highlight that the
solution to the system
is a pair of values (in this case, pounds of raisins and walnuts) that meet both constraints. This means the pair of values is a solution to both equations. Graphing is an effective way to see the solution to both equations, if one exists.


correct_option_index: 0